КОМПЛЕКСНИЙ АНАЛІЗ ВЛАСТИВОСТЕЙ КОДУ ХЕММІНГА В СУПЕРПОЗИЦІЇ ТЕОРІЙ КОДУВАННЯ, ЗАВАДОСТІЙКОСТІ ТА ІНФОРМАЦІЇ

Abstract

In the modern era, the task of transferring and preserving information is extremely important. Errors can often occur when data is transmitted over channels with interference or when it is stored on an unreliable medium. Scientific studies of the properties of electrical signals, which are information carriers in electronic communication systems, traditionally have three directions: the theory of immunity, the theory of information, and the theory of probability. Each of these theories has its own historical lineage, outstanding achievements and untapped potential. But the peculiarity of these studies is that more often they are implemented independently of each other, using their own set of initial parameters, criteria and performance indicators, quantitative characteristics. The article examines the properties of Hamming codes, which have been actively used in practice for more than 70 years, in the context of determining their most effective area of use in various fields. The specific interest in Hamming codes is also caused by the fact that they have a unique property: they are - according to the classical definition - perfect codes. The difficulty lies in the fact that the concept of "the most effective area of use" in each of the mentioned theories is special, therefore, a comprehensive analysis of the properties of the Hamming code in the superposition of the theories of coding, immunity and information is a relevant and interesting research task. From the standpoint of coding theory, Hamming codes have the same correcting ability - they correct only one error in a block. After all, as the length of the block n increases, the redundancy decreases. At the same time, the degree of closeness of the Hamming codes to the Plotkin limit increases. Therefore, preference according to these criteria should be given to Hamming codes with the largest block length n. From the standpoint of the theory of immunity, the best is the code that provides the best correction indicators at the smallest value of the data transmission channel parameter h2 (signal/noise ratio) under the conditions of ensuring the required reliability. Compared to the case of no coding, the shortest code – the Hamming code (7,4) – gives the greatest gain. From the point of view of information theory, the transmission of signals using Hamming codes requires speed correction (acceleration) in the communication channel in order to prevent the loss of source symbols due to the redundancy of the correction code. At the same time, the advantage of the Hamming code (7,4) over other, longer codes remains. According to the integral indicator - the energy costs to achieve the required reliability without losing the speed of information transmission from the source - preference should be given to the Hamming code with parameters (7, 4) with a gain in the signal-to-noise ratio Δh2 = 2.72 dB compared to the case without coding.